author | wenzelm |
Wed, 23 Jul 2025 14:53:21 +0200 | |
changeset 82898 | 89da4dcd1fa8 |
parent 76216 | 9fc34f76b4e8 |
permissions | -rw-r--r-- |
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(* Title: ZF/AC/AC18_AC19.thy |
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Author: Krzysztof Grabczewski |
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The proof of AC1 \<Longrightarrow> AC18 \<Longrightarrow> AC19 \<Longrightarrow> AC1 |
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*) |
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theory AC18_AC19 |
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imports AC_Equiv |
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begin |
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definition |
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uu :: "i \<Rightarrow> i" where |
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"uu(a) \<equiv> {c \<union> {0}. c \<in> a}" |
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(* ********************************************************************** *) |
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(* AC1 \<Longrightarrow> AC18 *) |
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(* ********************************************************************** *) |
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lemma PROD_subsets: |
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"\<lbrakk>f \<in> (\<Prod>b \<in> {P(a). a \<in> A}. b); \<forall>a \<in> A. P(a)<=Q(a)\<rbrakk> |
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\<Longrightarrow> (\<lambda>a \<in> A. f`P(a)) \<in> (\<Prod>a \<in> A. Q(a))" |
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by (rule lam_type, drule apply_type, auto) |
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lemma lemma_AC18: |
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"\<lbrakk>\<forall>A. 0 \<notin> A \<longrightarrow> (\<exists>f. f \<in> (\<Prod>X \<in> A. X)); A \<noteq> 0\<rbrakk> |
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\<Longrightarrow> (\<Inter>a \<in> A. \<Union>b \<in> B(a). X(a, b)) \<subseteq> |
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(\<Union>f \<in> \<Prod>a \<in> A. B(a). \<Inter>a \<in> A. X(a, f`a))" |
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apply (rule subsetI) |
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apply (erule_tac x = "{{b \<in> B (a) . x \<in> X (a,b) }. a \<in> A}" in allE) |
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apply (erule impE, fast) |
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apply (erule exE) |
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apply (rule UN_I) |
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apply (fast elim!: PROD_subsets) |
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apply (simp, fast elim!: not_emptyE dest: apply_type [OF _ RepFunI]) |
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done |
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lemma AC1_AC18: "AC1 \<Longrightarrow> PROP AC18" |
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unfolding AC1_def |
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apply (rule AC18.intro) |
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apply (fast elim!: lemma_AC18 apply_type intro!: equalityI INT_I UN_I) |
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done |
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(* ********************************************************************** *) |
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(* AC18 \<Longrightarrow> AC19 *) |
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(* ********************************************************************** *) |
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theorem (in AC18) AC19 |
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unfolding AC19_def |
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apply (intro allI impI) |
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apply (rule AC18 [of _ "\<lambda>x. x", THEN mp], blast) |
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done |
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(* ********************************************************************** *) |
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(* AC19 \<Longrightarrow> AC1 *) |
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(* ********************************************************************** *) |
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lemma RepRep_conj: |
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"\<lbrakk>A \<noteq> 0; 0 \<notin> A\<rbrakk> \<Longrightarrow> {uu(a). a \<in> A} \<noteq> 0 \<and> 0 \<notin> {uu(a). a \<in> A}" |
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apply (unfold uu_def, auto) |
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apply (blast dest!: sym [THEN RepFun_eq_0_iff [THEN iffD1]]) |
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done |
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lemma lemma1_1: "\<lbrakk>c \<in> a; x = c \<union> {0}; x \<notin> a\<rbrakk> \<Longrightarrow> x - {0} \<in> a" |
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apply clarify |
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apply (rule subst_elem, assumption) |
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apply (fast elim: notE subst_elem) |
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done |
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lemma lemma1_2: |
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"\<lbrakk>f`(uu(a)) \<notin> a; f \<in> (\<Prod>B \<in> {uu(a). a \<in> A}. B); a \<in> A\<rbrakk> |
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\<Longrightarrow> f`(uu(a))-{0} \<in> a" |
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apply (unfold uu_def, fast elim!: lemma1_1 dest!: apply_type) |
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done |
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lemma lemma1: "\<exists>f. f \<in> (\<Prod>B \<in> {uu(a). a \<in> A}. B) \<Longrightarrow> \<exists>f. f \<in> (\<Prod>B \<in> A. B)" |
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apply (erule exE) |
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apply (rule_tac x = "\<lambda>a\<in>A. if (f` (uu(a)) \<in> a, f` (uu(a)), f` (uu(a))-{0})" |
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in exI) |
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apply (rule lam_type) |
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apply (simp add: lemma1_2) |
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done |
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lemma lemma2_1: "a\<noteq>0 \<Longrightarrow> 0 \<in> (\<Union>b \<in> uu(a). b)" |
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by (unfold uu_def, auto) |
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lemma lemma2: "\<lbrakk>A\<noteq>0; 0\<notin>A\<rbrakk> \<Longrightarrow> (\<Inter>x \<in> {uu(a). a \<in> A}. \<Union>b \<in> x. b) \<noteq> 0" |
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apply (erule not_emptyE) |
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apply (rule_tac a = 0 in not_emptyI) |
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apply (fast intro!: lemma2_1) |
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done |
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parents:
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lemma AC19_AC1: "AC19 \<Longrightarrow> AC1" |
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apply (unfold AC19_def AC1_def, clarify) |
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apply (case_tac "A=0", force) |
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apply (erule_tac x = "{uu (a) . a \<in> A}" in allE) |
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apply (erule impE) |
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apply (erule RepRep_conj, assumption) |
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apply (rule lemma1) |
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apply (drule lemma2, assumption, auto) |
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done |
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end |